$30.00

## Description

Problem 1. 1. Given two statements S and T : (a) Compare the truth tables of

^{• } ^{the} ^{negation } ^{of} ^{S} ^{∧} ^{T} ^{,}

and of

• (¬S) ∨ (¬T ).

What can you conclude?

(b) Give the truth table of ¬(S ⇒ T ). Then write ¬(S ⇒ T ) using ¬T .

2. Consider the following statement:

If it is raining then I will take the bus, and otherwise I will ride my bicycle.

(a) Convert the above statement into propositional calculus, using ∧, ∨, ¬, and =⇒ .

Be sure to define any statements P, Q, that you use. (b) Write the negation of the above statement,

i. with symbols

ii. in plain English.

Problem 2. Consider the following two sets of natural numbers.

A = {2x − 1 : x ∈ N} = {1, 3, 5, 7, 9, . . .}

B = {3x : x ∈ N} = {3, 6, 9, 12, 15, . . .}

Give a description of the following two sets. A list of the first ten elements followed by . . .

is sufficient.

1. {x ∈ N : (x ∈ A) or (x ∈ B)}

2. {x ∈ N : (x ∈ A) =⇒ (x ∈ B)}

3. {x ∈ N : (x ∈ B) =⇒ (x ∈ A)}

4. {x ∈ N : (x ∈ A =⇒ x ∈ B) and (x ∈ B =⇒ x ∈ A)}

Problem 3. For x ∈ R, prove the following statement:

If |x| > 10 then x^{2} + 40 > 14x.

Problem 4. Let a, b, and c be integers. Consider the statements:

P : c divides ab Q: c divides a R: c divides b

1. Write the statement P =⇒ Q ∨ R in words.

2. Give an example of integers a, b and c for which the statement in part 1. is false.

Problem 5. Let n ∈ Z. Prove the following claim:

If 4 divides n − 1, then n is odd and (−1)

^{−}

_{n} _{1}

^{2 } ^{=} ^{1}^{.}

Problem 6. 1. Let n ∈ Z. Prove that if 5n is even then n is even.

2. Let n ∈ Z. Prove that if 5 divides n and 2 divides n, then 10 divides n.

3. Is the following statement true?

For n ∈ Z, if 6 divides n and 2 divides n, then 12 divides n.